A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.

Prediction of structural integrity, robustness and service life using advanced finite element methods

Corrosion is a common phenomenon and critical aspects of steel structural application. It affects the daily design, inspection and maintenance in structural engineering, especially for the heavy and complex industrial applications, where the steel structures are subjected to hash corrosive environments in combination of high working stress condition and often in open field and/or under high temperature production environments. In the paper, it presents the actual engineering application of advanced finite element methods in the predication of the structural integrity and robustness at a designed service life for the furnaces of alumina production, which was operated in the high temperature, corrosive environments and rotating with high working stress condition.

Effects of surgical joint destabilization on load sharing between ligamentous structures in the thoracic spine : a finite element investigation

Background: In vitro investigations have demonstrated the importance of the ribcage in stabilising the thoracic spine. Surgical alterations of the ribcage may change load-sharing patterns in the thoracic spine. Computer models are used in this study to explore the effect of surgical disruption of the rib-vertebrae connections on ligament load-sharing in the thoracic spine. Methods: A finite element model of a T7-8 motion segment, including the T8 rib, was developed using CT-derived spinal anatomy for the Visible Woman. Both the intact motion segment and the motion segment with four successive stages of destabilization (discectomy and removal of right costovertebral joint, right costotransverse joint and left costovertebral joint) were analysed for a 2000Nmm moment in flexion/extension, lateral bending and axial rotation. Joint rotational moments were compared with existing in vitro data and a detailed investigation of the load sharing between the posterior ligaments carried out. Findings: The simulated motion segment demonstrated acceptable agreement with in vitro data at all stages of destabilization. Under lateral bending and axial rotation, the costovertebral joints were of critical importance in resisting applied moments. In comparison to the intact joint, anterior destabilization increases the total moment contributed by the posterior ligaments. Interpretation: Surgical removal of the costovertebral joints may lead to excessive rotational motion in a spinal joint, increasing the risk of overload and damage to the remaining ligaments. The findings of this study are particularly relevant for surgical procedures involving rib head resection, such as some techniques for scoliosis deformity correction.

Updating and verification for multi-scale finite element model of structure behavior

The method on concurrent multi-scale model of structural behavior (CMSM-of-SB) for the purpose of structural health monitoring including model updating and validating has been studied. The detailed process of model updating and validating is discussed in terms of reduced scale specimen of the steel box girder in longitudinal stiffening truss of a long span bridge. Firstly, some influence factors affecting the accuracy of the CMSM-of-SB including the boundary restraint regidity, the geometry and material parameters on the toe of the weld and its neighbor are analyzed using sensitivity method. Then, sensitivity-based model updating technology is adopted to update the developed CMSM-of-SB and model verification is carried out through calculating and comparing stresses on different locations under various loading from dynamic characteristic and static response. It can be concluded that the CMSM-of-SB based on the substructure method is valid.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Tooth fracture risk analysis based on a new finite element dental structure models using micro-CT data

The finite element (FE) analysis is an effective method to study the strength and predict the fracture risk of endodontically-treated teeth. This paper presents a rapid method developed to generate a comprehensive tooth FE model using data retrieved from micro-computed tomography (μCT). With this method, the inhomogeneity of material properties of teeth was included into the model without dividing the tooth model into different regions. The material properties of the tooth were assumed to be related to the mineral density. The fracture risk at different tooth portions was assessed for root canal treatments. The micro-CT images of a tooth were processed by a Matlab software programme and the CT numbers were retrieved. The tooth contours were obtained with thresholding segmentation using Amira. The inner and outer surfaces of the tooth were imported into Solidworks and a three-dimensional (3D) tooth model was constructed. An assembly of the tooth model with the periodontal ligament (PDL) layer and surrounding bone was imported into ABAQUS. The material properties of the tooth were calculated from the retrieved CT numbers via ABAQUS user's subroutines. Three root canal geometries (original and two enlargements) were investigated. The proposed method in this study can generate detailed 3D finite element models of a tooth with different root canal enlargements and filling materials, and would be very useful for the assessment of the fracture risk at different tooth portions after root canal treatments.

Finite element analyses of lipped channel beams with Web openings in shear

Cold-formed steel members are increasingly used as primary structural elements in buildings due to the availability of thin and high strength steels and advanced cold-forming technologies. Cold-formed lipped channel beams (LCB) are commonly used as flexural members such as floor joists and bearers. Shear behaviour of LCBs with web openings is more complicated and their shear capacities are considerably reduced by the presence of web openings. However, limited research has been undertaken on the shear behaviour and strength of LCBs with web openings. Hence a numerical study was undertaken to investigate the shear behaviour and strength of LCBs with web openings. Finite element models of simply supported LCBs with aspect ratios of 1.0 and 1.5 were considered under a mid-span load. They were then validated by comparing their results with test results and used in a detailed parametric study. Experimental and numerical results showed that the current design rules in cold-formed steel structures design codes are very conservative for the shear design of LCBs with web openings. Improved design equations were therefore proposed for the shear strength of LCBs with web openings. This paper presents the details of this numerical study of LCBs with web openings, and the results.

Finite element modelling of load bearing steel stud walls under real building fires

Fire safety has become an important part in structural design due to the ever increasing loss of properties and lives during fires. Conventionally the fire rating of load bearing wall systems made of Light gauge Steel Frames (LSF) is determined using fire tests based on the standard time-temperature curve given in ISO 834 (ISO, 1999). The standard time-temperature curve given in ISO 834 (ISO, 1999) originated from the application of wood burning furnaces in the early 1900s. However, modern commercial and residential buildings make use of thermoplastic materials, which mean considerably high fuel loads. Hence a detailed fire research study into the performance of LSF walls was undertaken using the developed real fire curves based on Eurocode parametric curves (ECS, 2002) and Barnett’s BFD curves (Barnett, 2002) using both full scale fire tests and numerical studies. It included LSF walls without any insulation, and the recently developed externally insulated composite panel system. This paper presents the details of the numerical studies and the results. It also includes brief details of the development of real building fire curves and experimental studies.

A least squares based finite volume method for the Cahn-Hilliard and Cahn-Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn-Hilliard (CH) and recently proposed Cahn-Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.

Ergodic capacity of the slotted amplify and forward relay channel with finite relays

Abstract—In this paper we investigate the capacity of a general class of the slotted amplify and forward (SAF) relaying protocol where multiple, though a finite number of relays may transmit in a given cooperative slot and the relay terminals being half-duplex have a finite slot memory capacity. We derive an expression for the capacity per channel use of this generalized SAF channel assuming all source to relay, relay to destination and source to destination channel gains are independent and modeled as complex Gaussian. We show through the analysis of eigenvalue distributions that the increase in limiting capacity per channel use is marginal with the increase of relay terminals.